# Boundary values, random walks, and $\ell^p$-cohomology in degree one

### Antoine Gournay

Université de Neuchâtel, Switzerland

## Abstract

The vanishing of reduced $\ell^2$-cohomology for amenable groups can be traced to the work of Cheeger and Gromov in [10]. The subject matter here is reduced $\ell^p$-cohomology for $p \in ]1,\infty[$, particularly its vanishing. Results for the triviality of $\underline {\ell^pH}^1(G)$ are obtained, for example: when $p \in ]1,2]$ and $G$ is amenable; when $p \in ]1,\infty[$ and $G$ is Liouville (e.g. of intermediate growth).

This is done by answering a question of Pansu in [34, §1.9] for graphs satisfying certain isoperimetric profile. Namely, the triviality of the reduced $\ell^p$-cohomology is equivalent to the absence of non-constant harmonic functions with gradient in $\ell^q$ ($q$ depends on the profile). In particular, one reduces questions of non-linear analysis ($p$-harmonic functions) to linear ones (harmonic functions with a very restrictive growth condition).

## Cite this article

Antoine Gournay, Boundary values, random walks, and $\ell^p$-cohomology in degree one. Groups Geom. Dyn. 9 (2015), no. 4, pp. 1153–1184

DOI 10.4171/GGD/337